Weak well-posedness and weak discretization error for stable-driven SDEs with Lebesgue drift

We are interested in the discretization of stable driven SDEs with additive noise for $\alpha$ $\in$ (1, 2) and Lq – Lp drift under the Serrin type condition $\alpha$/q + d/p < $\alpha$ – 1. We show weak existence and uniqueness as well as heat kernel estimates for the SDE and obtain a convergence rate of order (1/$\alpha$)*($\alpha$ – 1 – $\alpha$/q - d/p) for the difference of the densities for the Euler scheme approximation involving suitably cutoffed and time randomized drifts.